Whenever we accelerate we feel forces. We are flung to the side when we speed around a corner, we snap upwards when the plummet of a bungee jump is suddenly arrested, our arms fly out when we spin around.

Yet beneath these familiar observations is a profound puzzle about the nature of space and time: what does it mean to accelerate? Is acceleration relative, like motion at a constant velocity? Would an astronaut spinning alone in an empty universe experience the same forces? Einstein brilliantly recognized that acceleration and gravity were one and the same. But since gravity is the influence of matter, could acceleration too depend on matter, and thus be relative?

This tantalizing idea, known as Mach's principle, has seduced generations of physicists. Yet attempts to implement it have floundered. We will revisit it with the new insight that one must also take into account the influence of distant matter at the furthest reaches of space-time. We hope to demonstrate that an astronaut would feel exactly the same forces if, instead of him, it was this "boundary matter" that was spinning. Thus all motion would be relative; indeed, the very shape of space-time would be determined by matter.

The success of quantum information theory suggests that quantum theory is best understood as a noncommutative generalization of classical probability theory. However, the usual formalism of quantum theory is a closer analog of the theory of classical stochastic processes than of abstract probability in the sense of Kolmogorov. This project aims to investigate the extent to which such an abstract formulation is possible, in order to better understand the nature of information in quantum theory, and understand how to apply quantum theory in the absence of any background causal structure. The main ideas of the project are as follows. Firstly, survey the analogs of conditional probability that have been suggested for quantum theory and develop new ones if necessary. Secondly, study the extent to which the notion of a subsystem in quantum theory can be extended to collections of observables that do not necessarily commute. Thirdly, determine the equivalence classes causal relations that are compatible with the observed statistics in a quantum experiment, within the framework of a quantum generalization of network theory. Finally, understand how quantum probability can be made compatible with all the major interpretations of quantum theory, with particular emphasis on the subjective probability approach.